Mass transport induced by internal Kelvin waves
beneath shore fast ice
E. Støylen and J. E. Weber
Department of Geosciences, University of Oslo,
P.B. 1022 Blindern, N-0315 Oslo, Norway
Background and scope
Coastally trapped Internal Kelvin waves may reach
large amplitudes, and are always propagating in
one direction. When induced by tides the motion is
steady, motivating study of the second order wave
drift. This drift may yield systematic transport of
biologic material and pollutants in near-shore
regions. In this study we derive a mathematical
model for the second order wave drift, and compare
the results with numerical simulations in an
idealized basin.
Second order theory
Numerical simulation
The upper layer currents are integrated in the
vertical between material surfaces ξ and H1, in order
to retain the Lagrangian drift properties of the flow.
The resulting flux terms are separated in a
fluctuating (tilde) and a mean (overbar) part:
We utilize a linear vertically integrated reduced
gravity model in a circular geometry. We assume
known wave motion, and insert the resulting
radiation stress terms as geometrically dependent
forcing terms over the domain. The variables are the
Eulerian fluxes, and the mean interface
displacement.
~
U = U +U
The mean contribution is to second order in wave
steepness. It consists of the general Stokes drift in
addition to a mean Eulerian term dependent on
friction:
U = US +UE
Mathematical model
We look at a two-layer reduced gravity model under
ice, shown in Fig. 1. The lower layer is very deep,
densities ρ are constant and the dominant motion is
the upper layer currents u1 along the coast and v1
normal to the coast, and interface elevation ξ.
Friction enters the system through a thin boundary
layer δ under the ice. Letting the wave propagate
along positive x-axis with coast at y<0, the first
order internal wave motion is
~
ξ = Ae
−α x − y / a
The Stokes term is derived directly from the linear
wave. The vertically integrated equations are
averaged over a wave period to obtain the following
set of equations solving the Eulerian motion
(subscript indicates partial derivatives):
− fVE − c ξ = −(3 / 2)c1U Sx − cD U E U E / H
2
1 x
a)
fU E − c ξ y = 0
2
1
cos(kx + ly - ωt)
where α is a damping coefficient proportional to δ.
The phase lines are tilted due to the wave number
component l which is also dependent on the
friction.
2
1
Results are shown in Fig. 2. The black lines in a)
indicate the wave generation area, and max wave
amplitude is just to the right of this area. The
interface maximum is close to the wave entrance,
decreasing along the coast, and more abruptly
toward the interior of the basin. The currents
generally follow the isobars, trapped near the coast.
Close to the edge of the trapping region there is a
current component normal to the coast, in
accordance with theoretical results. Outside the
trapping area, a small return current is observable.
Realistic topographies
In order to study the internal Kelvin wave
generation, numerical model runs are made for a
two-layer system, solving the linear wave motion
from oscillatory interface boundary conditions. Fig.
3 a) show the result from a semi-enclosed
rectangular basin with a shallow and narrow sound
at x=40km, and open boundary at x=0. Fig. 3 b)
show result for a topography resembling the Van
Mijen fjord in Svalbard. The two-layer density
profile applied do not resemble the physical
conditions in this fjord, we merely compare results
from the two geometries to compare the resulting
wave patterns. The striking similarities indicate that
the internal Kelvin wave is a robust phenomenon
with respect to topographies.
a)
b)
U Ex + VEy = −U Sx
Here we assume that friction acts on the mean
motion through some drag coefficient cD. This
problem resembles the steady barotropic storm
surge problem; the forcing here is divergence of
Stokes flux (radiation stress). Also note a forcing
term in the continuity equation.
b)
The resulting Eulerian fluxes are
UE =
αH 1
cD
c1 Ae
−αx − y / a
 A

αH 1
−2 y / a
− 2αx
−y/a
−αx
+
1− e
1− e
VE = α ac1 A
e
e 
cD
 2 H1

(
)
(
)
Fig 2: Result from a reduced gravity model. Interface
displacement (a) and mean current (b), from the marked
region in a). Red in a) indicates low pressure areas (thin
upper layer thickness).
Fig 3: Result from two-layer model. Interface displacements from rectangular topography (a) and Van Mijen fjord
topography (b). Amplitudes are normalized with respect to
max amplitude A.
Conclusions
Fig 1: Sketch of the two-layer model. H1<<H2, constant
densities, and a rigid ice lid at z=H1. The under-ice
pressure Ps is not necessarily constant.
Inserting typical values valid for the Eastern Barents
sea, we obtain U E = 2.2U S , which is comparable to
Longuet-Higgins’ result U E = 3 / 2U S for laminar flow.
Interestingly, the flow normal to the coast is not zero
everywhere.
• Eulerian mean currents may be comparable to the conventional Stokes drift in magnitude.
• Due to damping of wave amplitude, a mean flow normal to the coast arises. This is verified numerically.
• Internal Kelvin wave propagation is not easily disturbed by topography, thus transport mechanisms from
these waves may be important in stratified coastal environments, e.g. the Barents Sea and Arctic fjords